The actual meaning of Mouthbreather is someone who has a deviated septum and therefore has trouble breathing through the nose. But people misunderstand the reason that they breathe through their mouths and misintepret it as an insult and don't realize that it is not their fault.

Poor John has a deviated septum and can't breathe very well through his nose. He doesn't have enough money or insurance for the surgery, so he has to endure all the insults and is labeled a mouthbreather

A person who breathes loudly through their mouth with a nasally voice. Generally seen playing Dungeons and Dragons in the school library and looking at pictures of naked anime women in the computer labs.
Studies indicate that mouth breathers also have loud obnoxious laughter.

The ugly guy with the long red hair who is taking an arts degree but likes to pretend he's in Computer Science.

Derogatory. Special species of fanboy dork who breathes through his mouth. When not collecting funny books, watching cartoons, playing dungeons and dragons, or video games, he is stalking goth girls or getting beat up.

The pimply faced mouthbreather with coke bottle glasses crept into the comic bookstore. He watched the goth girl for 5 minutes before he walked up and said to her, "Gee, (weez, weez, weez) you remind me of Electra from the X-Men funny books...(weez, weez, pant)